Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity
Jianhua Huang Xingfu Zou
Discrete & Continuous Dynamical Systems - A 2003, 9(4): 925-936 doi: 10.3934/dcds.2003.9.925
In this paper, we establish the existence of traveling wavefronts for delayed reaction diffusion systems without quasimonotonicity in the reaction term, by using Schauder's fixed point theorem. We show the merit of our result by applying it to the Belousov-Zhabotinskii reaction model with two delays.
keywords: Traveling wavefront reaction diffusion systems quasimonotone delay upper solution and lower solution. Schauder’s fixed point theorem
Dynamics of evolutionary competition between budding and lytic viral release strategies
Xiulan Lai Xingfu Zou
Mathematical Biosciences & Engineering 2014, 11(5): 1091-1113 doi: 10.3934/mbe.2014.11.1091
In this paper, we consider the evolutionary competition between budding and lytic viral release strategies, using a delay differential equation model with distributed delay. When antibody is not established, the dynamics of competition depends on the respective basic reproductive ratios of the two viruses. If the basic reproductive ratio of budding virus is greater than that of lytic virus and one, budding virus can survive. When antibody is established for both strains but the neutralization capacities are the same for both strains, consequence of the competition also depends only on the basic reproductive ratios of the budding and lytic viruses. Using two concrete forms of the viral production functions, we are also able to conclude that budding virus will outcompete if the rates of viral production, death rates of infected cells and neutralizing capacities of the antibodies are the same for budding and lytic viruses. In this case, budding strategy would have an evolutionary advantage. However, if the antibody neutralization capacity for the budding virus is larger than that for the lytic virus, the lytic virus can outcompete the budding virus provided that its reproductive ratio is very high. An explicit threshold is derived.
keywords: competition stability antibody infection age releasing strategy burst size. budding Virus dynamics lytic
On a reaction-diffusion model for sterile insect release method with release on the boundary
Xin Li Xingfu Zou
Discrete & Continuous Dynamical Systems - B 2012, 17(7): 2509-2522 doi: 10.3934/dcdsb.2012.17.2509
We consider a partial differential equation model that describes the sterile insect release method (SIRM) in a bounded 1-dimensional domain (interval). Unlike everywhere-releasing in the domain as considered in previous works [17] and [14] , we propose the mechanism of releasing on the boundary only. We show existence of the fertile-free steady state and prove its stability under some conditions. By using the upper-lower solution method, we also show that under some other conditions there may exist a coexistence steady state. Biological implications of our mathematical results are that the SIRM with releasing only on the boundary can successfully eradicate the fertile insects as long as the strength of the sterile releasing is reasonably large, while the method may also fail if the releasing is not sufficient.
keywords: diffusion steady state coexistence. upper- lower solution Sterile insect release method stability
On latencies in malaria infections and their impact on the disease dynamics
Yanyu Xiao Xingfu Zou
Mathematical Biosciences & Engineering 2013, 10(2): 463-481 doi: 10.3934/mbe.2013.10.463
In this paper, we modify the classic Ross-Macdonald model for malaria disease dynamics by incorporating latencies both for human beings and female mosquitoes. One novelty of our model is that we introduce two general probability functions ($P_1(t)$ and $P_2(t)$) to reflect the fact that the latencies differ from individuals to individuals. We justify the well-posedness of the new model, identify the basic reproduction number $\mathcal{R}_0$ for the model and analyze the dynamics of the model. We show that when $\mathcal{R}_0 <1$, the disease free equilibrium $E_0$ is globally asymptotically stable, meaning that the malaria disease will eventually die out; and if $\mathcal{R}_0 >1$, $E_0$ becomes unstable. When $\mathcal{R}_0 >1$, we consider two specific forms for $P_1(t)$ and $P_2(t)$: (i) $P_1(t)$ and $P_2(t)$ are both exponential functions; (ii) $P_1(t)$ and $P_2(t)$ are both step functions. For (i), the model reduces to an ODE system, and for (ii), the long term disease dynamics are governed by a DDE system. In both cases, we are able to show that when $\mathcal{R}_0>1$ then the disease will persist; moreover if there is no recovery ($\gamma_1=0$), then all admissible positive solutions will converge to the unique endemic equilibrium. A significant impact of the latencies is that they reduce the basic reproduction number, regardless of the forms of the distributions.
keywords: Lyapunov function/functional stability latency Malaria persistence. basic reproduction number delay
Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays
Zhaohui Yuan Xingfu Zou
Mathematical Biosciences & Engineering 2013, 10(2): 483-498 doi: 10.3934/mbe.2013.10.483
We consider a mathematical model that describes the interactions of the HIV virus, CD4 cells and CTLs within host, which is a modification of some existing models by incorporating (i) two distributed kernels reflecting the variance of time for virus to invade into cells and the variance of time for invaded virions to reproduce within cells; (ii) a nonlinear incidence function $f$ for virus infections, and (iii) a nonlinear removal rate function $h$ for infected cells. By constructing Lyapunov functionals and subtle estimates of the derivatives of these Lyapunov functionals, we shown that the model has the threshold dynamics: if the basic reproduction number (BRN) is less than or equal to one, then the infection free equilibrium is globally asymptotically stable, meaning that HIV virus will be cleared; whereas if the BRN is larger than one, then there exist an infected equilibrium which is globally asymptotically stable, implying that the HIV-1 infection will persist in the host and the viral concentration will approach a positive constant level. This together with the dependence/independence of the BRN on $f$ and $h$ reveals the effect of the adoption of these nonlinear functions.
keywords: CTLs global stability. non-linear infection rate delay HIV
Wave fronts in neuronal fields with nonlocal post-synaptic axonal connections and delayed nonlocal feedback connections
Felicia Maria G. Magpantay Xingfu Zou
Mathematical Biosciences & Engineering 2010, 7(2): 421-442 doi: 10.3934/mbe.2010.7.421
We consider a neuronal network model with both axonal connections (in the form of synaptic coupling) and delayed non-local feedback connections. The kernel in the feedback channel is assumed to be a standard non-local one, while for the kernel in the synaptic coupling, four types are considered. The main concern is the existence of travelling wave front. By employing the speed index function, we are able to obtain the existence of a travelling wave front for each of these four types within certain range of model parameters. We are also able to describe how the feedback coupling strength and the magnitude of the delay affect the wave speed. Some particular kernel functions for these four cases are chosen to numerically demonstrate the theoretical results.
keywords: speed index function Neuronal networks integro-differential equation spatially non-local travelling wave front. delay feedback
A 3/2 stability result for a regulated logistic growth model
Xianhua Tang Xingfu Zou
Discrete & Continuous Dynamical Systems - B 2002, 2(2): 265-278 doi: 10.3934/dcdsb.2002.2.265
A sufficient condition is established for globally asymptotic stability of the positive equilibrium of a regulated logistic growth model with a delay in the state feedback. The result improves some existing criteria for this model. It is in a form that is related to the number $3/2$ and the coupling strength, and thus, is comparable to the well-known $3/2$ condition for the uncontrolled delayed logistic equation. The comparison seems to suggest that the mechanism of the control in this model might be inappropriate and new mechanism should be introduced.
keywords: Logistic feedback regulation. global stability
Effects of superinfection and cost of immunity on host-parasite co-evolution
Liman Dai Xingfu Zou
Discrete & Continuous Dynamical Systems - B 2017, 22(3): 809-829 doi: 10.3934/dcdsb.2017040

In this paper, we investigate the cost of immunological up- regulation caused by infection in a between-host transmission dynamical model with superinfection. After introducing a mutant host to an existing model, we explore this problem in (A) monomorphic case and (B) dimorphic case. For (A), we assume that only strain 1 parasite can infect the mutant host. We identify an appropriate fitness for the invasion of the mutant host by analyzing the local stability of the mutant free equilibrium. After specifying a trade-off between the production and recovery rates of infected hosts, we employ the adaptive dynamical approach to analyze the evolutionary and convergence stabilities of the corresponding singular strategy, leading to some conditions for continuously stable strategy, evolutionary branching point and repeller. For (B), a new fitness is introduced to measure the invasion of mutant host under the assumption that both parasite strains can infect the mutant host. By considering two trade-off functions, we can study the conditions for evolutionary stability, isoclinic stability and absolute convergence stability of the singular strategy. Our results show that the host evolution would not favour high degree of immunological up-regulation; moreover, superinfection would help the parasite with weaker virulence persist in hosts.

keywords: Dimorphic adaptive dynamics mutant co-evolution superinfection up-regulation trade-off function
Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay
Huiyan Zhu Xingfu Zou
Discrete & Continuous Dynamical Systems - B 2009, 12(2): 511-524 doi: 10.3934/dcdsb.2009.12.511
In this paper, we consider a mathematical model for HIV-1 infection with intracellular delay and cell-mediated immune response. A novel feature is that both cytotoxic T lymphocytes (CTLs) and the intracellular delay are incorporated into the model. We obtain a necessary and sufficient condition for the global stability of the infection-free equilibrium and give sufficient conditions for the local stability of the two infection equilibria: one without CTLs being activated and the other with. We also perform some numerical simulations which support the obtained theoretical results. These results show that larger intracellular delay may help eradicate the virus, while the activation of CTLs can only help reduce the virus load and increase the healthy CD$_4^+$ cells population in the long term sense.
keywords: Stability. intracellular delay immune response HIV-1 infection
Pattern formation of a predator-prey model with the cost of anti-predator behaviors
Xiaoying Wang Xingfu Zou
Mathematical Biosciences & Engineering 2018, 15(3): 775-805 doi: 10.3934/mbe.2018035

We propose and analyse a reaction-diffusion-advection predator-prey model in which we assume that predators move randomly but prey avoid predation by perceiving a repulsion along predator density gradient. Based on recent experimental evidence that anti-predator behaviors alone lead to a 40% reduction on prey reproduction rate, we also incorporate the cost of anti-predator responses into the local reaction terms in the model. Sufficient and necessary conditions of spatial pattern formation are obtained for various functional responses between prey and predators. By mathematical and numerical analyses, we find that small prey sensitivity to predation risk may lead to pattern formation if the Holling type Ⅱ functional response or the Beddington-DeAngelis functional response is adopted while large cost of anti-predator behaviors homogenises the system by excluding pattern formation. However, the ratio-dependent functional response gives an opposite result where large predator-taxis may lead to pattern formation but small cost of anti-predator behaviors inhibits the emergence of spatial heterogeneous solutions.

keywords: Pattern formation stability predator-prey model anti-predator behaviors bifurcation global stability

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